# Properties

 Label 2.112896.8t5.a.a Dimension 2 Group $Q_8$ Conductor $2^{8} \cdot 3^{2} \cdot 7^{2}$ Root number 1 Frobenius-Schur indicator -1

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## Basic invariants

 Dimension: $2$ Group: $Q_8$ Conductor: $112896= 2^{8} \cdot 3^{2} \cdot 7^{2}$ Artin number field: Splitting field of 8.8.1438916737499136.1 defined by $f= x^{8} - 84 x^{6} + 2268 x^{4} - 19404 x^{2} + 441$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $Q_8$ Parity: Even Determinant: 1.1.1t1.a.a Projective image: $C_2^2$ Projective field: Galois closure of $$\Q(\sqrt{3}, \sqrt{14})$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 47 }$ to precision 12.
Roots:
 $r_{ 1 }$ $=$ $4 + 30\cdot 47 + 2\cdot 47^{2} + 10\cdot 47^{3} + 24\cdot 47^{4} + 17\cdot 47^{5} + 34\cdot 47^{6} + 40\cdot 47^{7} + 41\cdot 47^{8} + 24\cdot 47^{9} + 7\cdot 47^{10} + 29\cdot 47^{11} +O\left(47^{ 12 }\right)$ $r_{ 2 }$ $=$ $10 + 45\cdot 47 + 24\cdot 47^{2} + 25\cdot 47^{3} + 3\cdot 47^{5} + 6\cdot 47^{6} + 42\cdot 47^{7} + 45\cdot 47^{8} + 41\cdot 47^{9} + 19\cdot 47^{10} + 4\cdot 47^{11} +O\left(47^{ 12 }\right)$ $r_{ 3 }$ $=$ $14 + 5\cdot 47 + 3\cdot 47^{2} + 6\cdot 47^{3} + 17\cdot 47^{4} + 45\cdot 47^{5} + 2\cdot 47^{6} + 32\cdot 47^{8} + 8\cdot 47^{9} + 41\cdot 47^{10} + 39\cdot 47^{11} +O\left(47^{ 12 }\right)$ $r_{ 4 }$ $=$ $17 + 27\cdot 47 + 42\cdot 47^{2} + 25\cdot 47^{3} + 10\cdot 47^{4} + 40\cdot 47^{5} + 7\cdot 47^{6} + 37\cdot 47^{7} + 10\cdot 47^{8} + 27\cdot 47^{9} + 2\cdot 47^{10} + 27\cdot 47^{11} +O\left(47^{ 12 }\right)$ $r_{ 5 }$ $=$ $30 + 19\cdot 47 + 4\cdot 47^{2} + 21\cdot 47^{3} + 36\cdot 47^{4} + 6\cdot 47^{5} + 39\cdot 47^{6} + 9\cdot 47^{7} + 36\cdot 47^{8} + 19\cdot 47^{9} + 44\cdot 47^{10} + 19\cdot 47^{11} +O\left(47^{ 12 }\right)$ $r_{ 6 }$ $=$ $33 + 41\cdot 47 + 43\cdot 47^{2} + 40\cdot 47^{3} + 29\cdot 47^{4} + 47^{5} + 44\cdot 47^{6} + 46\cdot 47^{7} + 14\cdot 47^{8} + 38\cdot 47^{9} + 5\cdot 47^{10} + 7\cdot 47^{11} +O\left(47^{ 12 }\right)$ $r_{ 7 }$ $=$ $37 + 47 + 22\cdot 47^{2} + 21\cdot 47^{3} + 46\cdot 47^{4} + 43\cdot 47^{5} + 40\cdot 47^{6} + 4\cdot 47^{7} + 47^{8} + 5\cdot 47^{9} + 27\cdot 47^{10} + 42\cdot 47^{11} +O\left(47^{ 12 }\right)$ $r_{ 8 }$ $=$ $43 + 16\cdot 47 + 44\cdot 47^{2} + 36\cdot 47^{3} + 22\cdot 47^{4} + 29\cdot 47^{5} + 12\cdot 47^{6} + 6\cdot 47^{7} + 5\cdot 47^{8} + 22\cdot 47^{9} + 39\cdot 47^{10} + 17\cdot 47^{11} +O\left(47^{ 12 }\right)$

### Generators of the action on the roots $r_1, \ldots, r_{ 8 }$

 Cycle notation $(1,8)(2,7)(3,6)(4,5)$ $(1,4,8,5)(2,6,7,3)$ $(1,2,8,7)(3,5,6,4)$

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 8 }$ Character value $1$ $1$ $()$ $2$ $1$ $2$ $(1,8)(2,7)(3,6)(4,5)$ $-2$ $2$ $4$ $(1,4,8,5)(2,6,7,3)$ $0$ $2$ $4$ $(1,2,8,7)(3,5,6,4)$ $0$ $2$ $4$ $(1,3,8,6)(2,4,7,5)$ $0$
The blue line marks the conjugacy class containing complex conjugation.